Trigonometry Examples

${(3 (\cos (48) + i \sin (48)))}^{5}$

Step 1

Apply the distributive property.

${(3 \cos (48) + 3 (i \sin (48)))}^{5}$

Step 2

Use the Binomial Theorem.

${(3 \cos (48))}^{5} + 5 {(3 \cos (48))}^{4} (3 i \sin (48)) + 10 {(3 \cos (48))}^{3} {(3 i \sin (48))}^{2} + 10 {(3 \cos (48))}^{2} {(3 i \sin (48))}^{3} + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Apply the product rule to $3 \cos (48)$ .

$3^{5} \cos^{5} (48) + 5 {(3 \cos (48))}^{4} (3 i \sin (48)) + 10 {(3 \cos (48))}^{3} {(3 i \sin (48))}^{2} + 10 {(3 \cos (48))}^{2} {(3 i \sin (48))}^{3} + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.2

Raise $3$ to the power of $5$ .

$243 \cos^{5} (48) + 5 {(3 \cos (48))}^{4} (3 i \sin (48)) + 10 {(3 \cos (48))}^{3} {(3 i \sin (48))}^{2} + 10 {(3 \cos (48))}^{2} {(3 i \sin (48))}^{3} + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.3

Apply the product rule to $3 \cos (48)$ .

$243 \cos^{5} (48) + 5 (3^{4} \cos^{4} (48)) (3 i \sin (48)) + 10 {(3 \cos (48))}^{3} {(3 i \sin (48))}^{2} + 10 {(3 \cos (48))}^{2} {(3 i \sin (48))}^{3} + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.4

Multiply $3^{4}$ by $3$ by adding the exponents.

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Step 3.1.4.1

Move $3$ .

$243 \cos^{5} (48) + 5 (3 \cdot 3^{4} \cos^{4} (48)) (i \sin (48)) + 10 {(3 \cos (48))}^{3} {(3 i \sin (48))}^{2} + 10 {(3 \cos (48))}^{2} {(3 i \sin (48))}^{3} + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.4.2

Multiply $3$ by $3^{4}$ .

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Step 3.1.4.2.1

Raise $3$ to the power of $1$ .

$243 \cos^{5} (48) + 5 (3^{1} \cdot 3^{4} \cos^{4} (48)) (i \sin (48)) + 10 {(3 \cos (48))}^{3} {(3 i \sin (48))}^{2} + 10 {(3 \cos (48))}^{2} {(3 i \sin (48))}^{3} + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$243 \cos^{5} (48) + 5 (3^{1 + 4} \cos^{4} (48)) (i \sin (48)) + 10 {(3 \cos (48))}^{3} {(3 i \sin (48))}^{2} + 10 {(3 \cos (48))}^{2} {(3 i \sin (48))}^{3} + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.4.3

Add $1$ and $4$ .

$243 \cos^{5} (48) + 5 (3^{5} \cos^{4} (48)) (i \sin (48)) + 10 {(3 \cos (48))}^{3} {(3 i \sin (48))}^{2} + 10 {(3 \cos (48))}^{2} {(3 i \sin (48))}^{3} + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.5

Raise $3$ to the power of $5$ .

$243 \cos^{5} (48) + 5 (243 \cos^{4} (48)) (i \sin (48)) + 10 {(3 \cos (48))}^{3} {(3 i \sin (48))}^{2} + 10 {(3 \cos (48))}^{2} {(3 i \sin (48))}^{3} + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.6

Multiply $243$ by $5$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) (i \sin (48)) + 10 {(3 \cos (48))}^{3} {(3 i \sin (48))}^{2} + 10 {(3 \cos (48))}^{2} {(3 i \sin (48))}^{3} + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.7

Apply the product rule to $3 \cos (48)$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) + 10 (3^{3} \cos^{3} (48)) {(3 i \sin (48))}^{2} + 10 {(3 \cos (48))}^{2} {(3 i \sin (48))}^{3} + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.8

Raise $3$ to the power of $3$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) + 10 (27 \cos^{3} (48)) {(3 i \sin (48))}^{2} + 10 {(3 \cos (48))}^{2} {(3 i \sin (48))}^{3} + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.9

Multiply $27$ by $10$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) + 270 \cos^{3} (48) {(3 i \sin (48))}^{2} + 10 {(3 \cos (48))}^{2} {(3 i \sin (48))}^{3} + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.10

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1.10.1

Apply the product rule to $3 i \sin (48)$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) + 270 \cos^{3} (48) ({(3 i)}^{2} \sin^{2} (48)) + 10 {(3 \cos (48))}^{2} {(3 i \sin (48))}^{3} + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.10.2

Apply the product rule to $3 i$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) + 270 \cos^{3} (48) (3^{2} i^{2} \sin^{2} (48)) + 10 {(3 \cos (48))}^{2} {(3 i \sin (48))}^{3} + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.11

Raise $3$ to the power of $2$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) + 270 \cos^{3} (48) (9 i^{2} \sin^{2} (48)) + 10 {(3 \cos (48))}^{2} {(3 i \sin (48))}^{3} + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.12

Rewrite $i^{2}$ as $- 1$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) + 270 \cos^{3} (48) (9 \cdot - 1 \sin^{2} (48)) + 10 {(3 \cos (48))}^{2} {(3 i \sin (48))}^{3} + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.13

Multiply $9$ by $- 1$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) + 270 \cos^{3} (48) (- 9 \sin^{2} (48)) + 10 {(3 \cos (48))}^{2} {(3 i \sin (48))}^{3} + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.14

Multiply $- 9$ by $270$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) + 10 {(3 \cos (48))}^{2} {(3 i \sin (48))}^{3} + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.15

Apply the product rule to $3 \cos (48)$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) + 10 (3^{2} \cos^{2} (48)) {(3 i \sin (48))}^{3} + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.16

Raise $3$ to the power of $2$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) + 10 (9 \cos^{2} (48)) {(3 i \sin (48))}^{3} + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.17

Multiply $9$ by $10$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) + 90 \cos^{2} (48) {(3 i \sin (48))}^{3} + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.18

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1.18.1

Apply the product rule to $3 i \sin (48)$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) + 90 \cos^{2} (48) ({(3 i)}^{3} \sin^{3} (48)) + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.18.2

Apply the product rule to $3 i$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) + 90 \cos^{2} (48) (3^{3} i^{3} \sin^{3} (48)) + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.19

Raise $3$ to the power of $3$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) + 90 \cos^{2} (48) (27 i^{3} \sin^{3} (48)) + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.20

Factor out $i^{2}$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) + 90 \cos^{2} (48) (27 (i^{2} \cdot i) \sin^{3} (48)) + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.21

Rewrite $i^{2}$ as $- 1$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) + 90 \cos^{2} (48) (27 (- 1 \cdot i) \sin^{3} (48)) + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.22

Rewrite $- 1 i$ as $- i$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) + 90 \cos^{2} (48) (27 (- i) \sin^{3} (48)) + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.23

Multiply $- 1$ by $27$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) + 90 \cos^{2} (48) (- 27 i \sin^{3} (48)) + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.24

Multiply $- 27$ by $90$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) - 2430 \cos^{2} (48) (i \sin^{3} (48)) + 5 (3 \cos (48)) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.25

Multiply $3$ by $5$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) - 2430 \cos^{2} (48) i \sin^{3} (48) + 15 \cos (48) {(3 i \sin (48))}^{4} + {(3 i \sin (48))}^{5}$

Step 3.1.26

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1.26.1

Apply the product rule to $3 i \sin (48)$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) - 2430 \cos^{2} (48) i \sin^{3} (48) + 15 \cos (48) ({(3 i)}^{4} \sin^{4} (48)) + {(3 i \sin (48))}^{5}$

Step 3.1.26.2

Apply the product rule to $3 i$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) - 2430 \cos^{2} (48) i \sin^{3} (48) + 15 \cos (48) (3^{4} i^{4} \sin^{4} (48)) + {(3 i \sin (48))}^{5}$

Step 3.1.27

Raise $3$ to the power of $4$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) - 2430 \cos^{2} (48) i \sin^{3} (48) + 15 \cos (48) (81 i^{4} \sin^{4} (48)) + {(3 i \sin (48))}^{5}$

Step 3.1.28

Rewrite $i^{4}$ as $1$ .

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Step 3.1.28.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) - 2430 \cos^{2} (48) i \sin^{3} (48) + 15 \cos (48) (81 {(i^{2})}^{2} \sin^{4} (48)) + {(3 i \sin (48))}^{5}$

Step 3.1.28.2

Rewrite $i^{2}$ as $- 1$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) - 2430 \cos^{2} (48) i \sin^{3} (48) + 15 \cos (48) (81 {(- 1)}^{2} \sin^{4} (48)) + {(3 i \sin (48))}^{5}$

Step 3.1.28.3

Raise $- 1$ to the power of $2$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) - 2430 \cos^{2} (48) i \sin^{3} (48) + 15 \cos (48) (81 \cdot 1 \sin^{4} (48)) + {(3 i \sin (48))}^{5}$

Step 3.1.29

Multiply $81$ by $1$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) - 2430 \cos^{2} (48) i \sin^{3} (48) + 15 \cos (48) (81 \sin^{4} (48)) + {(3 i \sin (48))}^{5}$

Step 3.1.30

Multiply $81$ by $15$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) - 2430 \cos^{2} (48) i \sin^{3} (48) + 1215 \cos (48) \sin^{4} (48) + {(3 i \sin (48))}^{5}$

Step 3.1.31

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1.31.1

Apply the product rule to $3 i \sin (48)$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) - 2430 \cos^{2} (48) i \sin^{3} (48) + 1215 \cos (48) \sin^{4} (48) + {(3 i)}^{5} \sin^{5} (48)$

Step 3.1.31.2

Apply the product rule to $3 i$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) - 2430 \cos^{2} (48) i \sin^{3} (48) + 1215 \cos (48) \sin^{4} (48) + 3^{5} i^{5} \sin^{5} (48)$

Step 3.1.32

Raise $3$ to the power of $5$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) - 2430 \cos^{2} (48) i \sin^{3} (48) + 1215 \cos (48) \sin^{4} (48) + 243 i^{5} \sin^{5} (48)$

Step 3.1.33

Factor out $i^{4}$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) - 2430 \cos^{2} (48) i \sin^{3} (48) + 1215 \cos (48) \sin^{4} (48) + 243 (i^{4} i) \sin^{5} (48)$

Step 3.1.34

Rewrite $i^{4}$ as $1$ .

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Step 3.1.34.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) - 2430 \cos^{2} (48) i \sin^{3} (48) + 1215 \cos (48) \sin^{4} (48) + 243 ({(i^{2})}^{2} i) \sin^{5} (48)$

Step 3.1.34.2

Rewrite $i^{2}$ as $- 1$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) - 2430 \cos^{2} (48) i \sin^{3} (48) + 1215 \cos (48) \sin^{4} (48) + 243 ({(- 1)}^{2} i) \sin^{5} (48)$

Step 3.1.34.3

Raise $- 1$ to the power of $2$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) - 2430 \cos^{2} (48) i \sin^{3} (48) + 1215 \cos (48) \sin^{4} (48) + 243 (1 i) \sin^{5} (48)$

Step 3.1.35

Multiply $i$ by $1$ .

$243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) - 2430 \cos^{2} (48) i \sin^{3} (48) + 1215 \cos (48) \sin^{4} (48) + 243 i \sin^{5} (48)$

Step 3.2

Simplify the expression.

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Step 3.2.1

Reorder factors in $243 \cos^{5} (48) + 1215 \cos^{4} (48) i \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) - 2430 \cos^{2} (48) i \sin^{3} (48) + 1215 \cos (48) \sin^{4} (48) + 243 i \sin^{5} (48)$ .

$243 \cos^{5} (48) + 1215 i \cos^{4} (48) \sin (48) - 2430 \cos^{3} (48) \sin^{2} (48) - 2430 i \cos^{2} (48) \sin^{3} (48) + 1215 \cos (48) \sin^{4} (48) + 243 i \sin^{5} (48)$

Step 3.2.2

Reorder $243 \cos^{5} (48) + 1215 i \cos^{4} (48) \sin (48)$ and $- 2430 \cos^{3} (48) \sin^{2} (48)$ .

$- 2430 \cos^{3} (48) \sin^{2} (48) + 243 \cos^{5} (48) + 1215 i \cos^{4} (48) \sin (48) - 2430 i \cos^{2} (48) \sin^{3} (48) + 1215 \cos (48) \sin^{4} (48) + 243 i \sin^{5} (48)$

Step 3.2.3

Reorder $- 2430 \cos^{3} (48) \sin^{2} (48) + 243 \cos^{5} (48) + 1215 i \cos^{4} (48) \sin (48) - 2430 i \cos^{2} (48) \sin^{3} (48)$ and $1215 \cos (48) \sin^{4} (48)$ .

$1215 \cos (48) \sin^{4} (48) - 2430 \cos^{3} (48) \sin^{2} (48) + 243 \cos^{5} (48) + 1215 i \cos^{4} (48) \sin (48) - 2430 i \cos^{2} (48) \sin^{3} (48) + 243 i \sin^{5} (48)$